Question

$\int\frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx $ is equal to :
(where $c$ is a constant of integration)

• A. $2x + sin \,x + 2 sin\,2x + c$
• B. $x + 2\,sinx + 2\,sin2x + c$
• C. $x + 2\, sin x + sin \,2x + c$
• D. $2x + sinx + sin2x + c$

Answer 1

Answer: (C)

$\int\frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx = \int \frac{2\sin \frac{5x}{2} \cos \frac{x}{2}}{2\sin \frac{x}{2} \cos \frac{x}{2}} dx $
$ =\int \frac{\sin 3x +\sin 2x}{\sin x} dx $
$ = \int \frac{3\sin x-4\sin^{3}x-2\sin x\cos x}{\sin x}dx $
$ = \int \left(3-4\sin^{2}x+2\cos x \right)dx $
$ =\int \left(3-2\left(1-\cos2x\right)+2\cos x\right)dx $
$ =\int\left(1+2\cos2x+2\cos x\right)dx $
$ =x+\sin2x+2\sin x +c $